We'll keep your hypothesis.
Make a word this way :
Do you agree that there are letters a ?
Plus, there are letters b (I'll let you do the counting).
Therefore, this word has letters.
So we're looking for all words of letters and containing k letters a and n-1 letters b.
This is the part you have to understand : it's equivalent to finding the possible places where b can be.
So this is the combinations of (p-1) among k+(n-1)..
If I have time, I'll try to post an example...